Univariate Analysis
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Raw Specific Gravity Box-and-Whiskers Plot
Bivariate Analysis
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Molecular Mass vs Specific Gravity Goossens Data
Bivariate Analysis
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Molecular Mass vs Boiling Point Goossens Data
Bivariate Analysis
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Specific Gravity vs Boiling Point Goossens Data
Test vs. Train
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Molecular Mass vs Boiling Point. Both Datasets
Test vs. Train
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Molecular Mass vs Specific Gravity. Both Datasets
Existing Correlations
| \(M_w\) |
Apparent Molecular Mass |
| \(T_b\) |
True Boiling Point Temperature |
| \(\gamma_o\) |
Specific Gravity |
| \(a_{00}..a_{09}\) |
Empirical Constants |
| \(K_w\) |
Characterization Factor (intermediate value) |
| \(X_0...X_3\) |
Intermediate Variables |
Existing Correlations
Hariu & Sage (1969)
\[
M_w = a_{00} + a_{01} K_w + a_{02} K_w^2 + a_{03} T_b K_w + a_04 T_b K_w^2 + a_{05} T_b^2 K_w + a_{06} T_b^2 K_w^2
\]
\[K_w =\frac{\sqrt[3]T_b}{\gamma_o}\]
Kesler & Lee (1976)
\[M_w = X_0 + \frac{X_1}{T_b} + \frac{X_2}{T_b^2}\]
\[X_0 = a_{00} + a_{01} γ_o+ \left (a_{02} + a_{03} γ_o \right ) T_b\]
\[
X_1 = \left (1+ a_{04} γ_o + a_{05}γ_o^2 \right ) \left (a_{06} + \frac{a_ {07}}{T_b} \right ) \cdot 10^7
\]
\[
X_2 = \left (1+ a_{08} γ_o+ a_{09} γ_o^2 \right ) \left (a_{10} + \frac{a_{11}}{T_b} \right ) \cdot 10^{12}
\]
American Petroleum Institute (1977)
\[
M_w = a_{00} e^ {\left (a_{01} T_b \right )} e^{\left (a_{02} γ_o \right )} T_b^{a_{03}} γ_o^ {a_{04}}
\]
Winn, Sim & Daubert (1980)
\[M_w = a_{00} T_b^{a_ {01}} γ_o^{a_{02}}\]
Riazi & Daubert (1980)
\[M_w = a_{00} T_b^{a_ {01}}γ_o^{a_{02}}\]
Rao & Bardon (1985)
\[ln {M_w} = (a_{00} + a_{01} K_w) ln (\frac{T_b} {a_{02} + a_{03} K_w} )\]
Riazi & Daubert (1987)
\[
M_w = a_{00} T_b^{a_{01}} γ_o^{a_{02}} e^{\left (a_{03} T_b + a_{04} γ_o + a_{05} T_b γ \right )}
\]
Goossens (1996)
\[M_w = a_{00} T_b^{X_0}\]
\[
X_0 =\frac {a_{03} + a_{04} ln {\left (\frac{T_b} {a_{05} - T_b} \right )}} {a_{01} γ_o + a_{02}}
\]
Linan (2011)
\[
M_w = a_{00} e^{\left (a_{01} T_b \right )} e^{\left (a_{02} γ_o \right )} T_b^ {a_{03}} γ_o^{a_{04}}
\]
Hosseinifar & Shahverdi (2021)
\[M_w = {\left [a_{00} T_b^{a_{01}} {\left (\frac{3+2γ_o} {3-γ_o} \right )}^{\frac{a_{02}}{2}} + a_{03} T_b^{a_{04}} {\left (\frac{3+2γ_o}{3-γ_o} \right )}^{\frac{a_{05}}{2}} \right ]}^{a_{06}}\]
Stratiev (2023)
\[
M_w = a_{00} + a_{01} e^{\left [a_{02} e^{\left (a_{03} \frac{T_b^{a_{06}}}{γ_o^{a_{05}}} \right )} \right ]}
\]
Operators to Consider
Partial List
| pow |
binary |
one expression raised to the power of another |
| log |
unary |
logarithm of an expression |
| exp |
unary |
antilogarithm of an expression |
| sqr |
unary |
expression squared |
| cub |
unary |
expression cubed |
| inv |
unary |
inverse of an expression |
Power Run
\[M_w=a_{00} \cdot a_{01}^{\gamma_o^{-a_{02}} + a_{03} \cdot T_b} + a_{04}\]
Correlation Coefficients
| Raw MW |
1.000000 |
0.997281 |
| This Equation |
0.997281 |
1.000000 |
Power Run
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Exponential Run
\[ M_w= - T_b \cdot \left(a_{00} \cdot T_b - a_{01}\right) \left(a_{02} \cdot 10^{-6} \left(a_{03} \cdot \gamma_o - 2 \cdot T_b \right) \left(T_b - a_{04}\right) - 1\right) + a_{05}\]
\[ M_w= a_{00}\cdot T_b + a_{01} \cdot e^{- \gamma_o^{2} + a_{02}\cdot \gamma_o + a_{03} \cdot T_b}\]
Correlation Coefficients
| Raw MW |
1.000000 |
0.997705 |
0.998420 |
| First Equation |
0.997705 |
1.000000 |
0.999497 |
| Second Equation |
0.998420 |
0.999497 |
1.000000 |
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Raw Data Scale Change
First Sparse Run
\[ M_w=-a_{00} \cdot \gamma_o +\frac{a_{01}\cdot \gamma_o}{T_b}+a_{02}\cdot T_b -a_{03}\]
Correlation Coefficients
| Raw Mw |
1.000000 |
0.970451 |
| This Equation |
0.970451 |
1.000000 |
First Sparse Run
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Second Sparse Run
\[ M_w=-a_{00}\cdot \gamma_o\cdot T_b +a_{01}\cdot T^2_b +a_{02}\cdot e^{\gamma_o} +a_{03}\]
Correlation Coefficients
| Raw Mw |
1.000000 |
|
| 1st Run Equation |
0.970451 |
|
| 2nd Run Equation |
0.985800 |
|
Second Sparse Run
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Validation Runs
| Raw Mw |
1.000000 |
| Default |
0.999957 |
| Goossens Correlation |
0.999939 |
| Power |
0.996964 |
| Exponential #2 |
0.998954 |
| Aeon #1 |
0.999973 |
| Aeon #2 |
0.999921 |
| Box-Cox |
0.999696 |
| Sparse #1 |
0.992303 |
| Sparse #2 |
0.996331 |
General Observations
- UOP Characterization Factor
- Replication
- Explainability
- Table Look-Up
\[ M_w^t = \left (\gamma_o \left(a_{00} T_b -a_{01} \right) log{(\gamma_o)} + a_{02} \right) log{(T_b)} \]
